AMS / MAA CLASSROOM RESOURCE MATERIALS VOL 44 44 Exploring Advanced Euclidean Geometry with Geogebra

AMS / MAA CLASSROOM RESOURCE MATERIALS VOL AMS / MAA CLASSROOM RESOURCE MATERIALS VOL 44 44 Exploring Advanced Euclidean Geometry with Geogebra

This book provides an inquiry-based introduction to advanced Euclidean geometry. It utilizes dynamic geometry software, specifically GeoGebra, to explore the statements and proofs of many of the most interesting theorems in the subject. Topics covered include triangle centers, inscribed, circumscribed, and escribed circles, medial and orthic triangles, the nine-point circle, duality, and the theorems of Ceva and Menelaus, as well as numerous applications of those theorems. The final chapter explores constructions in the Poincaré disk model for hyperbolic geometry. The book can be used either as a computer laboratory manual to supplement an undergraduate course in geometry or as a stand-alone introduction to advanced topics in Euclidean geometry. The text consists almost entirely of exercises (with hints) that guide students as they discover the geometric relationships for themselves. First the ideas are explored at the computer and then those ideas are assembled into a proof of the result under investigation. The goals are for the reader to experience the joy of discovering geometric relationships, to develop a deeper understanding of geometry, and to encourage an appreciation for the beauty of Euclidean geometry. |

Gerard A, Venema M A PRESS / MAA AMS

146 pages on 50lb stock • Trim Size 7 X 10 • Spine 5/16" 4-Color Process i i “EEG-master” — 2013/4/18 — 22:54 — page i — #1 i i

10.1090/clrm/044

Exploring Advanced Euclidean Geometry with GeoGebra

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c 2013 by the Mathematical Association of America, Inc.

Library of Congress Catalog Card Number 2013938569 Print edition ISBN 978-0-88385-784-7 Electronic edition ISBN 978-1-61444-111-3 Printed in the United States of America Current Printing (last digit): 10987654321

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Exploring Advanced Euclidean Geometry with GeoGebra

Gerard A. Venema Calvin College

Published and Distributed by The Mathematical Association of America

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Council on Publications and Communications Frank Farris, Chair Committee on Books Gerald M. Bryce, Chair Classroom Resource Materials Editorial Board Gerald M. Bryce, Editor Michael Bardzell Jennifer Bergner Diane L. Herrmann Paul R. Klingsberg Mary Morley Philip P. Mummert Mark Parker Barbara E. Reynolds Susan G. Staples Philip D. Strafﬁn Cynthia J Woodburn

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CLASSROOM RESOURCE MATERIALS

Classroom Resource Materials is intended to provide supplementary classroom material for students—laboratory exercises, projects, historical information, textbooks with unusual approaches for presenting mathematical ideas, career information, etc.

101 Careers in Mathematics, 2nd edition edited by Andrew Sterrett Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein Calculus: An Active Approach with Projects, Stephen Hilbert, Diane Driscoll Schwartz, Stan Seltzer, John Maceli, and Eric Robinson Calculus Mysteries and Thrillers, R. Grant Woods Conjecture and Proof, Miklós Laczkovich Counterexamples in Calculus, Sergiy Klymchuk Creative Mathematics, H. S. Wall Environmental Mathematics in the Classroom, edited by B. A. Fusaro and P. C. Kenschaft Excursions in Classical Analysis: Pathways to Advanced Problem Solving and Undergrad- uate Research, by Hongwei Chen Explorations in Complex Analysis, Michael A. Brilleslyper, Michael J. Dorff, Jane M. Mc- Dougall, James S. Rolf, Lisbeth E. Schaubroeck, Richard L. Stankewitz, and Kenneth Stephenson Exploratory Examples for Real Analysis, Joanne E. Snow and Kirk E. Weller Exploring Advanced Euclidean Geometry with GeoGebra, Gerard A. Venema Geometry From Africa: Mathematical and Educational Explorations, Paulus Gerdes Historical Modules for the Teaching and Learning of Mathematics (CD), edited by Victor Katz and Karen Dee Michalowicz Identification Numbers and Check Digit Schemes, Joseph Kirtland Interdisciplinary Lively Application Projects, edited by Chris Arney Inverse Problems: Activities for Undergraduates, Charles W. Groetsch Keeping it R.E.A.L.: Research Experiences for All Learners, Carla D. Martin and Anthony Tongen Laboratory Experiences in Group Theory, Ellen Maycock Parker Learn from the Masters, Frank Swetz, John Fauvel, Otto Bekken, Bengt Johansson, and Victor Katz Math Made Visual: Creating Images for Understanding Mathematics, Claudi Alsina and Roger B. Nelsen MathematicsGalore!: The First Five Years of the St. Marks Instituteof Mathematics, James Tanton Methods for Euclidean Geometry, Owen Byer, Felix Lazebnik, and Deirdre L. Smeltzer Ordinary Differential Equations: A Brief Eclectic Tour, David A. Sánchez Oval Track and Other Permutation Puzzles, John O. Kiltinen

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Paradoxes and Sophisms in Calculus, Sergiy Klymchuk and Susan Staples A Primer of Abstract Mathematics, Robert B. Ash Proofs Without Words, Roger B. Nelsen Proofs Without Words II, Roger B. Nelsen Rediscovering Mathematics: You Do the Math, Shai Simonson She Does Math!, edited by Marla Parker Solve This: Math Activities for Students and Clubs, James S. Tanton Student Manual for Mathematics for Business Decisions Part 1: Probability and Simula- tion, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Student Manual for Mathematics for Business Decisions Part 2: Calculus and Optimiza- tion, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Teaching Statistics Using Baseball, Jim Albert Visual Group Theory, Nathan C. Carter Which Numbers are Real?, Michael Henle Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go, An- nalisa Crannell, Gavin LaRose, Thomas Ratliff, and Elyn Rykken

MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789

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Preface

This book provides an inquiry-based introduction to advanced Euclidean geometry. It can be used either as a computer laboratory manual to supplement a course in the foundations of geometry or as a stand-alone introduction to advanced topics in Euclidean geometry. The geometric content is substantially the same as that of the ﬁrst half of the classic text Geometry Revisited by Coxeter and Greitzer [3]; the organization and method of study, however, are quite different. The book utilizes dynamic geometry software, speciﬁcally GeoGebra, to explore the statements and proofs of many of the most interesting theorems in advanced Euclidean geometry. The text consists almost entirely of exercises that guide students as they discover the mathematics and then come to understand it for themselves.

Geometric content The geometry studied in this book is Euclidean geometry. Euclidean geometry is named for Euclid of Alexandria, who lived from approximately 325 BC until about 265 BC. The ancient Greeks developed geometry to a remarkably advanced level and Euclid did his work during the later stages of that development. He wrote a series of books, called the Elements, that organize and summarize the geometry of ancient Greece. Euclid’s Elements became by far the best known geometry text in history and Euclid’s name is universally associated with geometry as a result. Roughly speaking, elementary Euclidean geometry is the geometry that is contained in Euclid’s writings. Most readers will already be familiar with a good bit of elementary Euclidean geometry since all of high school geometry falls into that category. Advanced Euclidean geometry is the geometry that was discovered later—it is geometry that was done after Euclid’s death but is still built on Euclid’s work. It is to be distinguished from non-Euclidean geometry, which is geometry based on axioms that are different from those used by Euclid. Throughout the centuries since Euclid lived, geometers have continued to develop Euclidean geometry and have discovered large numbers of interesting relationships. Their discoveries constitute advanced Euclidean geometry and are the subject matter of this text. Many of the results of advanced Euclidean geometry are quite surprising. Most people who study them for the ﬁrst time ﬁnd the theorems to be amazing, almost miraculous, and

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viii Preface

value them for their aesthetic appeal as much as for their utility. I hope that users of this book will come to appreciate the elegance and beauty of Euclidean geometry and better understand why the subject has captivated the interest of so many people over the past two thousand years. The book includes a study of the Poincar´edisk model for hyperbolic geometry. Since thismodel is built withinEuclidean geometry, it is an appropriatetopicfor studyina course on Euclidean geometry. Euclidean constructions, mostly utilizing inversions in circles, are used to illustrate many of the standard results of hyperbolic geometry.

Computer software This is not the kind of textbook that neatly lays out all the facts you should know about advanced Euclidean geometry. Instead, it is meant to be a guide to the subject that leads you to discover both the theorems and their proofs for yourself. To fully appreciate the geometry presented here, it is essential that you be actively involved in the exploration and discovery process. Do not read the book passively, but diligently work through the explorations yourself as you read them. The main tool used to facilitate active involvement and discovery is the software package GeoGebra. It enables users to explore the theorems of advanced Euclidean geometry, to discover many of the results for themselves, and to see the remarkable relationships with their own eyes. The book consists mostly of exercises, tied together by short explanations. The user of the book should work through all the exercises while reading the book. That way he or she will be guided through the discovery process. Any exercise that is marked with a star (*) is meant to be worked on a computer, using GeoGebra, while the remaining exercises should be worked using pencil and paper. No prior knowledge of GeoGebra is assumed; complete instructions on how to use GeoGebra are included in Chapters 1 and 3. GeoGebra is open source software that can be obtained free of charge from the website www.geogebra.org. That the software is free is important because it means that every student can have a copy. I believe it is essential that all students experience the discovery of geometric relationships for themselves. When expensive software packages are used, there is often only a limited number of copies available and not every student has access to one. Every student can have GeoGebra available all the time. One of the best features of GeoGebra is how easy it is to use. Even a beginner can quicklyproduce intricate diagrams that illustratecomplicated geometric relationships. Users soon learn to make useful tools that automate parts of the constructions. To ensure that every user of this book has the opportunity to experience that first hand, the reader is ex- pected to produce essentially all the diagrams and illustrations. For that reason the number of figures in the text is kept to a minimum and no disk containing professionally-produced GeoGebra documents is supplied with the book. GeoGebra is rapidly becoming the most popular and most widely used dynamic software package for geometry, but it is not the only one that can be used in conjunction with this text. Such programs as Geometer’s Sketchpad, Cabri Geometry, Cinderella, and Ge- ometry Expressions can also be utilized. The instructions that are included in Chapters 1 and 3 are specific to GeoGebra, but the rest of the book can be studied using any one of the programs mentioned.

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Preface ix

Proof A major accomplishment of the ancient Greeks was the introductionof logicand rigor into geometry. They proved their theorems from ﬁrst principles and thus their results are more certain and lasting than are mere observations from empirical data. The logical, deductive aspect of geometry is epitomized in Euclid’s Elements and proof continuesto be one of the hallmarks of geometry to this day. Until recently, all those who worked on advanced Euclidean geometry followed in Euclid’s footsteps and did geometry by proving theorems, using only pencil and paper. Now that computer programs such as GeoGebra are available as tools, we must reexamine the place of proof in geometry. Some might expect the use of dynamic software to displace the deductive approach to geometry, but there is no reason the two approaches cannot enhance each other. I hope this book will demonstrate that proof and computer exploration can coexist comfortably in geometry and that each can support the other. The exercises in this book will guide the student to use GeoGebra to explore and discover the statements of the theorems and then will go on to use GeoGebra to better understand the proofs of the theorems as well. At the end of thisprocess of discovery the student shouldbe able towritea proofof theresult thathas been discovered. In thisway the student will come to understand the material to a depth that would not be possible if just computer explorationor just pencil and paper proof were used and should come to appreciate the fact that proof is an integral part of exploration, discovery, and understanding in mathematics. Not only is proof an important part of the process by which we come to discover and understand geometric results, but the proofs also have a subtle beauty of their own. I hope that the experience of writing the proofs will help students to appreciate this aesthetic aspect of the subject as well. In this text the word “verify” will be used to describe the kind of conﬁrmation that is possible with GeoGebra. Thus to verify that the angle sum of a triangle is 180ı will mean to use GeoGebra to construct a triangle, measure its three angles, calculate the sum of the measures, and then to observe that GeoGebra reports that the sum is always equal to 180ı regardless of how the size and shape of the triangle are changed. On the other hand, to prove that the angle sum is 180ı will mean to supply a written logical argument based on the axioms and previously proved theorems of Euclidean geometry.

Two ways to use this book This book can be used as a manual for a computer laboratory that supplements a course in the foundations of geometry. The notation and terminology used here are consistent with The Foundations of Geometry [11], but this manual is designed to be used alongside any textbook on axiomatic geometry. The review chapter that is included at the beginning of the book establishes all the necessary terminology and notation. A class that meets for one three-hour computer lab session per week should be able to lightly cover most of the text in one semester. When the book is used as a lab manual, Chapter 0 is not covered separately, but serves as a reference for notation, terminology,and statements of theorems from elementary Euclidean geometry. Most of the other chapters can be covered in one laboratory session each. The exceptions are Chapters 6 and 10, which are quite short and could be combined, and Chapter 11, which will require two or three sessions to cover completely.

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x Preface

A course that emphasizes Euclidean geometry exclusively will omit Chapter 14 and probably Chapter 13 as well, since the main purpose of Chapter 13 is to develop the tools that are needed for Chapter 14. On the other hand, most instructors who are teaching a course that covers non-Euclidean geometry will want to cover the last chapter; to do so it will probably be necessary to omit many of the applications of the Theorem of Menelaus. A thorough coverage of Chapter 14 will require more than one session. At each lab session the instructor should assign an appropriate number of GeoGebra exercises, determined by the background of the students and the length of the laboratory session. It should be possible for students to read the short explanations during the session and work through the exercises on their own. A limited number of the written proofs can be assigned as homework following the lab session. A second way in which to use the book is as a text for an inquiry-based course in advanced Euclidean geometry. Such a course would be taught in a modiﬁed Moore style in which the instructor does almost no lecturing, but students work out the proofs for themselves. A course based on these notes would differ from other Moore-style courses in the use of computer software to facilitate the discovery and proof phases of the process. Another difference between this course and the traditional Moore-style course is that students should be encouraged to discuss the results of their GeoGebra explorations with each other. Class time is used for student computer exploration and student presentations of so- lutions to exercises. The notes break down the proofs into steps of manageable size and offer students numerous hints. It is my experience that the hints and suggestions offered are sufﬁcient to allow students to construct their own proofs of the theorems. The Geo- Gebra explorations form an integral part of the process of discovering the proof as well as the statement of the theorem. This second type of course would cover the entire book, including Chapter 0 and all the exercises in all chapters.

The preparation of teachers The basic recommendation in The Mathematical Education of Teachers [2] is that future teachers take courses that develop a deep understanding of the mathematics they will teach. There are many ways in which to achieve depth in geometry. One way, for example, is to understand what lies beneath high school geometry. This is accomplished by studying the foundations of geometry, by examining the assumptions that lie at the heart of the subject, and by understandinghow the results of the subject are built on those assumptions. Another way in which to achieve depth is to investigate what is built on top of the geometry that is included in the high school curriculum. That is what this course is designed to do. One direct beneﬁt of this course to future high school mathematics teachers is that those who take the course will develop facility in the use of GeoGebra. Dynamic geometry software such as GeoGebra will undoubtedly become much more common in the high school classroom in the future, so future teachers need to know how to use it and what it can do. In addition, software such as GeoGebra will likely lead to a revival of interest in advanced Euclidean geometry. When students learn to use GeoGebra they will have the capability to investigate geometric relationships that are more intricate than those studied in the traditional high school geometry course. A teacher who knows some advanced Euclidean geometry will have a store of interesting geometric results that can be used to motivate and excite students.

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Preface xi

Do it yourself! The philosophy of these notes is that students can and should work out the geometry for themselves. But students will soon discover that many of the GeoGebra tools they are asked to make in the exercises can be found on the world wide web. I believe students should be encouraged to make use of the mathematical resources available on the web, but that they also benefit from the experience of making the tools for themselves. Downloading a tool that someone else has made and using it is too passive an activity. Working through the constructions for themselves and seeing how the intricate constructions of advanced Euclidean geometry are based on the simple constructions from high school geometry will enable them to achieve a much deeper understanding than they would if they simply used ready-made tools. I believe it is especially important that future high school mathematics teachers have the experience of doing the constructions for themselves. Onlyin thisway do they come to know that they can truly understand mathematics for themselves and that they do not have to rely on others to work it out for them. The question of whether or not to rely on tools made by others comes up most especially in the last chapter. There are numerous high-quality tools available on the web that can be used to perform constructions in the Poincarédisk. Nonetheless I think students should work through the constructions for themselves so that they clearly understand how the hyperbolic constructionsare built on Euclidean ones. After they have built rudimentary tools of their own, they might want to find more polished tools on the web and add those to their toolboxes.

Acknowledgments I want tothank all those who helped me develop thismanuscript. Numerous Calvin College students and my colleague Chris Moseley gave useful feedback. Gerald Bryce and the following members of the MAA’s Classroom Resource Materials Editorial Board read the manuscript carefully and offered many valuable suggestions: Michael Bardzell, Salisbury University; Diane Hermann, University of Chicago; Phil Mummert, Taylor University; Phil Strafﬁn, Beloit College; Susan Staples, Texas Christian University; Cynthia Woodburn, Pittsburgh State University; and Holly Zullo, Carroll College. I also thank the members of the MAA publications department, especially Carol Baxter and Beverly Ruedi, for their help and for making the production process go smoothly. Finally, I thank my wife Patricia whose patient support is essential to everything I do.

Gerard A. Venema [email protected] April, 2013

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Contents

Preface vii 0 A Quick Review of Elementary Euclidean Geometry 1 0.1 Measurementandcongruence...... 1 0.2 Angleaddition...... 2 0.3 Trianglesandtrianglecongruenceconditions ...... 3 0.4 Separationandcontinuity ...... 4 0.5 Theexteriorangletheorem ...... 5 0.6 Perpendicularlinesandparallellines ...... 5 0.7 ThePythagoreantheorem ...... 7 0.8 Similartriangles...... 8 0.9 Quadrilaterals ...... 9 0.10Circlesandinscribedangles ...... 10 0.11Area ...... 11 1 The Elements of GeoGebra 13 1.1 Gettingstarted:theGeoGebratoolbar...... 13 1.2 Simpleconstructionsandthedragtest ...... 16 1.3 Measurementandcalculation ...... 18 1.4 Enhancingthesketch ...... 20 2 The Classical Triangle Centers 23 2.1 Concurrentlines...... 23 2.2 Mediansandthecentroid ...... 24 2.3 Altitudesandtheorthocenter ...... 25 2.4 Perpendicularbisectorsand thecircumcenter ...... 26 2.5 TheEulerline ...... 27 3 Advanced Techniques in GeoGebra 31 3.1 User-deﬁnedtools...... 31 3.2 Checkboxes...... 33 3.3 ThePythagoreantheoremrevisited ...... 34

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xiv Contents

4 Circumscribed, Inscribed, and Escribed Circles 39 4.1 The circumscribed circle and the circumcenter ...... 39 4.2 Theinscribedcircleandtheincenter ...... 41 4.3 Theescribedcirclesandtheexcenters ...... 42 4.4 TheGergonnepointandtheNagelpoint ...... 43 4.5 Heron’sformula...... 44 5 The Medial and Orthic Triangles 47 5.1 Themedialtriangle ...... 47 5.2 Theorthictriangle...... 48 5.3 Ceviantriangles ...... 50 5.4 Pedaltriangles...... 51 6 Quadrilaterals 53 6.1 Basicdeﬁnitions...... 53 6.2 Convexandcrossedquadrilaterals ...... 54 6.3 Cyclicquadrilaterals...... 55 6.4 Diagonals ...... 56 7 The Nine-Point Circle 57 7.1 Thenine-pointcircle...... 57 7.2 Thenine-pointcenter ...... 59 7.3 Feuerbach’stheorem...... 60 8 Ceva’s Theorem 63 8.1 ExploringCeva’stheorem...... 63 8.2 Sensedratiosandidealpoints ...... 65 8.3 ThestandardformofCeva’stheorem ...... 68 8.4 ThetrigonometricformofCeva’stheorem ...... 71 8.5 Theconcurrencetheorems...... 72 8.6 Isotomic and isogonal conjugates and the symmedian point...... 73 9 The Theorem of Menelaus 77 9.1 Duality...... 77 9.2 ThetheoremofMenelaus ...... 78 10 Circles and Lines 81 10.1Thepowerofapoint...... 81 10.2Theradicalaxis ...... 83 10.3Theradicalcenter ...... 84 11 Applications of the Theorem of Menelaus 85 11.1Tangentlinesandanglebisectors ...... 85 11.2Desargues’theorem ...... 86 11.3Pascal’smystichexagram ...... 88 11.4Brianchon’stheorem...... 90 11.5Pappus’stheorem ...... 91 11.6Simson’stheorem ...... 93 11.7Ptolemy’stheorem...... 96

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Contents xv

11.8Thebutterflytheorem ...... 97 12 Additional Topics in Triangle Geometry 99 12.1Napoleon’stheoremandtheNapoleonpoint ...... 99 12.2TheTorricellipoint ...... 100 12.3vanAubel’stheorem...... 100 12.4Miquel’stheoremandMiquelpoints ...... 101 12.5TheFermatpoint ...... 101 12.6Morley’stheorem ...... 102 13 Inversions in Circles 105 13.1Invertingpoints ...... 105 13.2Invertingcirclesandlines ...... 107 13.3Othogonality...... 108 13.4Anglesanddistances ...... 110 14 The PoincaréDisk 111 14.1 The Poincarédisk model for hyperbolicgeometry ...... 111 14.2Thehyperbolicstraightedge...... 113 14.3Commonperpendiculars...... 114 14.4Thehyperboliccompass...... 116 14.5Otherhyperbolictools...... 117 14.6Trianglecentersinhyperbolicgeometry ...... 118 References 121 Index 123 About the Author 129

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References

[1] Claudi Alsina and Roger B. Nelsen. Icons of Mathematics. The Dolciani Mathematical Expo- sitions, volume 45. The Mathematical Association of America, Washington, DC, 2011. [2] Conference Board of the Mathematical Sciences CBMS. The Mathematical Education of Teachers. Issues in Mathematics Education, volume 11. The Mathematical Association of America and the American Mathematical Society, Washington, DC, 2001. [3] H. S. M. Coxeter and S. L. Greitzer. Geometry Revisited. New Mathematical Library, volume 19. The Mathematical Association of America, Washington, DC, 1967. [4] Dana Densmore, editor. The Bones: A Handy, Where-to-find-it Pocket Reference Companion to Euclid’s Elements. Green Lion Press, Santa Fe, New Mexico, 2002. [5] Dana Densmore, editor. Euclid’s Elements. Green Lion Press, Santa Fe, New Mexico, 2002. [6] William Dunham. Euler: The Master of Us All. The Dolciani Mathematical Expositions, volume 22. The Mathematical Association of America, Washington, DC, 1999. [7] Chaim Goodman-Strauss. Compass and straightedge in the Poincarédisk. American Mathe- matical Monthly, 108:38–49, 2001. [8] Sir ThomasL. Heath. The Thirteen Books of Euclid’s Elements with Introduction and Commen- tary. Dover Publications, Inc., Mineola, New York, 1956. [9] I. Martin Isaacs. Geometry for College Students. The Brooks/Cole Series in Advanced Mathe- matics. Brooks/Cole, Pacific Grove, CA, 2001. [10] Michael McDaniel. The polygons are all right. preprint, 2006. [11] Gerard A. Venema. The Foundations of Geometry. Pearson Education, Inc., Boston, second edition, 2012.

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Index

*-d exercises, viii, 13 in GeoGebra, 18 Cartesian plane, 111 acute angle, 2 center additivity of area, 12 of a circle, 10 advanced Euclidean geometry, vii of a square, 100 alternate interior angles theorem, 6 of mass, 25 altitude, 25 central angle, 10 concurrence, 72 central angle theorem, 10 altitude concurrence theorem, 26 centroid, 24, 72 angle, 2 hyperbolic, 118 angle addition postulate, 2 Ceva’s theorem angle bisector, 4 standard form, 51, 68 concurrence, 73 trigonometric form, 71 angle bisector concurrence theorem, 41 Ceva, G., 50, 63 angle sum theorem, 7 Cevian angle-angle-side theorem, 4 line, 50, 63 angle-side-angle theorem, 3 proper, 63 anticomplementary triangle, 48 triangle, 50 antimedial, 48 check boxes, 33 arc intercepted by, 10 Cinderella, viii Archimedes, 44 circle, 10 area, 11 circle-circle continuity, 5 in GeoGebra, 19 circle-line continuity, 5 Beltrami, E., 111 circumcenter, 27, 39, 72 betweenness hyperbolic, 118 for points, 1 circumcircle, 39 for rays, 2 hyperbolic, 118 bisect, 20 circumradius, 39 Brahmagupta’s formula, 56 circumscribed Brianchon’s theorem, 90, 91 about a circle, 90 Brianchon, C., 57, 78, 90 circle, 39 bride’s chair, 35 classical triangle center, 23 butterﬂy theorem, 97 coaxial, 86 concave quadrilateral, 54 Cabri Geometry, viii concurrence theorems, 72 calculation concurrency problem, 63

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124 Index

concurrent, 23 I.47, 35 congruent I.5, 4 angles, 2 I.8, 3 quadrilaterals, 9, 53 III.20, 10 segments, 1 III.21, 10 triangles, 3 III.22, 55 conjugate, 73 converse, 56 convex quadrilateral, 54 III.3, 11 copolar, 86 III.31, 11 corresponding angles, 6 converse, 11 corresponding angles theorem, 7 III.32, 81 corresponding central angle, 10 III.36, 81 create a tool, 31 VI.2, 8 cross-ratio, 110 VI.31, 35 crossbar theorem, 5, 69 VI.4, 8 crossed quadrilateral, 54 VI.6, 8 cyclic quadrilateral, 55 Euclidean geometry, vii Eudoxus, 8 de Longchampspoint, 48 Euler line, 28, 60 dependent objects, 15 hyperbolic, 119 Desargues’s theorem, 86 Euler line theorem, 28 Desargues, G., 77, 86 Euler, L., 23, 27, 57 diagonal excenter, 42, 73 of a quadrilateral, 9, 56 excircle, 42 diameter extended of a circle, 10 Euclidean plane, 66 directed extended law of sines, 40, 52 angle measure, 71 exterior angle, 5 distance, 66 exterior angle theorem, 5 distance, 1 external tangents theorem, 11 drag test, 17 drop a perpendicular, 6 Fermat point, 101 duality, 77 Fermat, P., 101 Feuerbach elementary Euclidean geometry, 1 point, 60 equicircle, 42 triangle, 60 equilateral triangle, 16 Feuerbach’s theorem, 60 escribed circle, 42 Feuerbach,K. W., 57 Euclid, vii foot Euclid’s Elements, vii, 1 of a perpendicular,6 Euclid’s ﬁfth postulate, 6 of an altitude, 25 Euclid’s Proposition free objects, 15 I.1, 16 I.12, 6 general position, 96 I.15, 3 GeoGebra, viii, 13 I.16, 5 area, 19 I.26, 3 calculation, 18 I.27, 6 check boxes, 33 I.28, 7 create a tool, 31 I.29, 6 measuring tools, 18 I.32, 7 save a tool, 32 I.34, 9 toolbar, 13 I.4, 3 user-deﬁned tools, 31

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Index 125

Geometer’s Sketchpad, viii medial triangle, 47 Gergonne point, 43, 73 hyperbolic, 118 Gergonne, J., 44 median, 24 concurrence, 72 Heron, 44 hyperbolic, 118 hexagon, 88 median concurrence theorem, 24 hyperbolic Menelaus of Alexandria, 77 compass, 116 Menelaus point, 78 geometry, 111 proper, 78 parallel postulate, 112 Menelaus’s theorem straightedge, 113 standard form, 51, 79 hypotenuse-leg theorem, 4 trigonometric form, 80 midpoint, 7 ideal point, 66 quadrilateral, 20, 55 image of a set, 107 Miquel point, 101 incenter, 42, 73 Miquel’s theorem, 101 hyperbolic, 118 Miquel, A., 101 incident, 77 model for a geometry, 111 incircle, 41, 42 Morley triangle, 102 hyperbolic, 118 Morley’s theorem, 103 inradius, 42 Morley, F., 103 inscribed movable point, 15 angle, 10 Move tool, 14 circle, 41, 42 hexagon, 88 Nagel point, 44, 73 quadrilateral, 55 Nagel, C., 44 inscribed angle theorem, 10, 56 Napoleon interior point, 100 angle, 5 triangle, 100 of an angle, 2 Napoleon Bonaparte, 99 internal angle bisector, 29 Napoleon’s theorem, 100 inversion in a circle, 105 neutral geometry, 112 inversive plane, 105 New Point tool, 14 involution, 73 nine-point isogonal center, 59 conjugate, 74 circle, 57 of a Cevian line, 74 nine-point center theorem, 60 isosceles triangle theorem, 4 nine-point circle theorem, 57 isotomic conjugate, 73 non-Euclidean geometry, vii, 111 Jordan curve theorem, 54 obtuse angle, 2 Lehmus, C., 29 opposite sides of a quadrilateral, 9, 53 length of a segment, 1 Options menu, 13 line, 1 ordinary point, 66 at inﬁnity, 66 orthic triangle, 48 Line tools, 14 hyperbolic, 119 more, 16 orthocenter, 26, 72 linear pair, 2 hyperbolic, 119 linear pair theorem, 2 orthogonal circles, 108

mark a point, 14 Pappus of Alexandria, 91 measure of an angle, 2 Pappus’s theorem, 92 measuring tools, 18 parallel, 6

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126 Index

parallel postulate, 6, 111 radical axis theorem, 84 parallel projection theorem, 8 radical center theorem, 84 parallelogram, 9, 54, 56 radius Pascal line, 88 of a circle, 10 Pascal’s mystic hexagram, 88 ray, 1 Pascal, B., 78, 88 rectangle, 9, 56 Pasch’s axiom, 4 region, 11 Pasch, M., 4 rhombus, 9, 56 pedal right angle, 2 line, 94 triangle, 51 SAS similarity criterion, 8 perpendicular, 6 save a tool, 32 perpendicular bisector, 7 secant line theorem, 11 concurrence, 72 second pedal triangle, 51 perspective segment, 1 from a line, 86 semiperimeter, 44 from a point, 86 sensed ratio, 66 perspector, 86 side perspectrix, 86 of a polygon, 11 Playfair’s postulate, 6 of a quadrilateral, 9, 53 Poincarédisk model, 111 of a triangle, 3, 25 Poincaré, H., 111 side-angle-side theorem, 3 point side-side-side theorem, 4 at infinity, 66, 105 sideline, 25 of concurrency, 23 similar triangles, 8 of perspective, 86 similar triangles theorem, 8 Point tools Simson line, 52, 94 other, 14 Simson’s theorem, 52, 93 pointwise characterization Simson, R., 93 of angle bisector, 4 square, 9, 56 of perpendicular bisector, 7 standard form pole of Ceva’s theorem, 68 of a Simson line, 94 of Menelaus’s theorem, 79 polygon, 11 Steiner, J., 29 Polygon tools, 16 Steiner-Lehmus theorem, 29 Poncelet, J., 57 supplements, 2 power of a point, 81, 82 symmedian, 76 projective geometry, 77 point, 76 proper synthetic, 111 Cevian line, 63 Menelaus point, 78 tangent line, 10, 85 prove, ix tangent line theorem, 10 Ptolemy, 96 Thales’ theorem, 11 Ptolemy’s theorem, 96 third pedal triangle, 51 Pythagoras, 7 toolbar, 13 Pythagorean theorem, 8, 34 topology, 54 Torricelli point, 100 quadrangle, 53 Torricelli, E., 100, 101 quadrilateral, 9, 53 transformation, 105 transversal, 6 radical trapezoid, 9 axis, 83 triangle, 3 center, 84 center, 23

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Index 127

hyperbolic, 118 congruence conditions, 3 triangle congruence condition AAS, 4 ASA, 3 HL, 4 SAS, 3 SSS, 4 trigonometric form of Ceva’s theorem, 71 of Menelaus’s theorem, 80 trilateral, 53, 78 trisect, 102 tritangent circle, 42

user-deﬁned tools, 31

van Aubel’s theorem, 100 Varignon’s theorem, 55 Varignon, P., 55 Vecten, 35 Vecten conﬁguration, 35 Vecten point, 36, 73 verify, ix vertex of a polygon, 11 of a quadrilateral, 9, 53 of a triangle, 3 vertical angles, 2 pair, 2 vertical angles theorem, 3 View menu, 13

Wallace, W., 93

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About the Author

Gerard Venema earned an A.B. in mathematics from Calvin College and a Ph.D. from the University of Utah. After completing his education he spent two years in a postdoc- toral position at the University of Texas at Austin and another two years as a Member of the Institute for Advanced Study in Princeton, NJ. He then returned to his alma mater, Calvin College, and has been a faculty member there ever since. While on the Calvin Col- lege faculty he has also held visiting faculty positions at the University of Tennessee, the University of Michigan, and Michigan State University. He spent two years as Program Di- rector for Topology, Geometry, and Foundations in the Division of Mathematical Sciences at the National Science Foundation. Venema is a member of the American Mathematical Society and the Mathematical Association of America. He served for ten years as an Associate Editor of the American Mathematical Monthly and currently sits on the editorial board of MAA FOCUS. Venema has served the Michigan Section of the MAA as chair and is the 2013 recipient of the section’s distinguished service award. He currently holds the position of MAA Associate Secretary and is a member of the Association’s Board of Governors. Venema is the author of two other books. One is an undergraduate textbook, Founda- tions of Geometry, published by Pearson Education, Inc., which is now in its second edition. The other is a research monograph, Embeddings in Manifolds, coauthored by Robert J. Daverman, that was published by the American Mathematical Society as volume 106 in its Graduate Studies in Mathematics series. In addition to the books, Venema is author of over thirty research articles in geometric topology.

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i i AMS / MAA CLASSROOM RESOURCE MATERIALS VOL AMS / MAA CLASSROOM RESOURCE MATERIALS VOL 44 44 Exploring Advanced Euclidean Geometry with Geogebra

Gerard A, Venema M A PRESS / MAA AMS

146 pages on 50lb stock • Trim Size 7 X 10 • Spine 5/16" 4-Color Process